In this work we present two independent inverse problem methods. In the first chapter we address the problem of source localization. Localizing sources in physical systems represents a class of inverse problems with broad scientific and engineering applications. This chapter is concerned with the development of a non-iterative source sensitivity approach for the localization of sources in linear systems under steady-state. We show that our proposed approach can be applied to a broad class of physical problems, ranging from source localization in elastodynamics and acoustics to source detection in heat/mass transport problems. The source sensitivity field introduced in this work represents the change of a cost functional caused by the appearance of an infinitesimal source is a given domain (or its boundary). In order to extract macroscopic inferences, we apply a threshold to the source sensitivity field in a way that parallels the application of the topological derivative concept in shape identification. We establish precise formulas for the source sensitivity field using a direct approach and a Lagrangian formulation. We show that computing the source sensitivity field entails just obtaining the solution of a single adjoint problem. Hence, the computational expense of obtaining the source sensitivity is of the same order as that of solving one forward problem. We illustrate the performance of the method through numerical examples drawn from the areas of elastodynamics, acoustics, and heat and mass transport. Our results show that our proposed approach could be used on its own as a source detection tool or to obtain initial guesses for more quantitative iterative gradient-based minimization strategies. In the second chapter we focus on material characterization. Material identification is integral to medical imaging, finite element calibration, non destructive testing, and other engineering applications. We propose an iterative computational framework for nonlinear material identification with transient data. Our method centers on the weak enforcement of the internal force computation, through which we derive a modified internal force equation. We subsequently enforce potentially sparse measurements in a least squares penalty term. The modified internal force equation results in a fully space-time coupled forward and adjoint problem. We consider two steps at each iteration. First the solution to the coupled problem, and second the material parameter update. Our approach generalizes the technique used for linear elastic materials. For our numerical examples, we focus on the Iwan constitutive model, commonly used to model frictional interactions in mechanical joints. We show several numerical examples exploring the accuracy of the coupled problem solution as well as the material reconstruction. We conclude with larger examples requiring distributed computation in order to demonstrate not only the algorithmic properties, but the computational scalability.